Hillary Rodham Clinton campaigned at full speed across Manhattan yesterday, saying she doesn’t care if she ends up running against Mayor Giuliani – or “whoever the Republicans put up.”

Collecting a union endorsement, urging more federal funding at a breast-cancer luncheon and winding up the day at a Greek awards dinner, Clinton said she had no idea what Giuliani was up to – and wasn’t giving it a second thought.

“Whatever is happening on the other side is not affecting my efforts,” Clinton told reporters after winning the backing of the 200,000-member Building Trades Council.

Reporters bombarded the first lady with questions about Giuliani and his still-uncertain candidacy at a press conference at a West Side construction site.

“I have no control over what the Republicans do,” Clinton said when asked if she would be disappointed if Giuliani bailed out.

“I’m just not thinking about that. That’s something other people will decide, not me and not my campaign.”

Clinton said she would continue to campaign on bread-and-butter issues like education, health care and the economy, regardless of who runs against her.

“In effect, I’m doing a job interview [with the voters] and whoever the Republicans put up is going to be doing a job interview,” she said.

Clinton aides said it would be a huge advantage if Giuliani withdraws, since recent polls have shown the first lady topping all other potential GOP rivals except Gov. Pataki, who has said he won’t run.

It would be difficult for any Republican newcomer to quickly assemble a campaign organization and fill a war chest to take on Clinton, who has raised more than $12 million and has a 10-month head start on the campaign trail.

Speaking of possible Giuliani substitute Rick Lazio, the Suffolk County congressman, one Clinton adviser said his inexperience as a statewide candidate could be deadly.

“He’s going to make mistakes,” the adviser said. “This is not that easy. Hillary found that out but now she’s doing better. Lazio would be facing the same thing, but he wouldn’t have the luxury of the time she had to get things right.”

Clinton will be formally nominated today at the state Democratic Convention in Albany to succeed the retiring Sen. Daniel Patrick Moynihan.

Aides said President Clinton would not attend the 5,000-person convention.